Besides, Kileel et al. [1]} prove a bottleneck property of deep polynomial networks. That is, a too narrow layer is a bottleneck and may “chock” the polynomial network such that the network can never fill the ambient space. The ambient space \(Sym_r(\mathbb {R}^d)\)  is the space of homogeneous polynomials of degree \(r\)  in \(d\)  variables. A polynomial network filling the ambient space satisfies \(\mathcal {F}_{m, r}=Sym_{r^{h-1}}(\mathbb {R}^d)^c\) . They show that, network architectures filling the ambient space can be helpful for optimization and training.
